Línies bàsiques de treball

Then, we have

Ve_x(t, η) =



where I_n is defined in (2.5.11). Hence,

|res₀| ≤ σ(t)Y

also provides the time-frequency zone for eVx_k(t, η). To describe those time-frequency zones mathematically, let 0 < τ₀ < 1 be a given small number and the threshold:

O_k = {(t, η) : |G_k

σ(t)(η − φ⁰_k(t)

| > τ₀, t ∈ R}. (2.6.6)

Assuming |G_k(ξ)| is even and decreasing for ξ ≥ 0, then we may write O_k as

O_k= {(t, η) : |η − φ⁰_k(t)| < α_k

σ(t), t ∈ R} (2.6.7) where α_k = α_k(t) is obtained by solving |G_k(ξ)| = τ₀.In this instance, we will say that multicomponent signal x(t) is well-separated and there is σ(t) such that

O_k∩ O_l = ∅, k 6= l.

We can use a Gaussian function defined in (2.5.24) as an example. For this g, we can obtain (see [15]),

G_k(u) = 1

p1 − i2πφ⁰⁰_k(t)σ²(t)e

− 2π²u²

1 + (2πφ⁰⁰_k(t)σ²(t))^2(1+i2πφ

00

k(t)σ²(t)))

.

Thus,

|G_k(u)| = 1

(1 + (2πφ⁰⁰_k(t)σ²(t))²) 1 4 e

− 2π²

1 + (2πφ⁰⁰_k(t)σ²(t))²) u².

The solution of |G_k(u) = τ₀| ⇔ e

− 2π²u²

1 + (2πφ⁰⁰_k(t)σ²(t))² = τ₀((1+(2πφ⁰⁰_k(t)σ²(t))²) 1 4 Therefore, in this case, assume τ₀((1 + (2πφ⁰⁰_k(t)σ²(t))²)

1 4 ≤ 1,

α_k = q

1 + (2πφ⁰⁰_k(t)σ²(t))²) 1 2π

r 2ln(1

τ₀) −1

2ln(1 + (2πφ⁰⁰_k(t)σ²(t))²)).

The main theorem on the 2nd-order adptive FSST can be written in more notations observe:

G_j,k(t, η) = Z

R

e^{i2π(φ0k(t)τ +12φ00k(t)τ2)} τ^j

σ(t)^j+1g( τ

σ(t))e^−i2πητdτ. (2.6.8)

= F

e^{iπ(φ00k(t)τ2)}τ^jg(τ )

σ(t)(η − φ⁰_k(t))
. (2.6.9)

Clearly,

G_0,k(t, η) = G_k σ(η − φ⁰_k(t))
,

and, when j ≥ 1, we can see that

G_j,k = 1

(−i2π)^jG^(j)_k σ(t)(η − φ⁰_k(t))
. (2.6.10) Let res₁, res₂, res⁰₀, and res⁰₁ be the residuals defined as res₀ in (2.6.2) with g(τ ) replaced respectively by g₁(τ ), g₂(τ ), g⁰(τ ), and g₃(τ ) = τ g⁰(τ ). We therefore have the estimates for these residuals that are similar to (2.6.4).

|res₁| ≤ σ(t)Y

1

(t), |res₂| ≤ σ(t)Y

2

(t), |res⁰₀| ≤ σ(t) fY

0(t), |res⁰₁| ≤ σ(t) fY

1(t),

where

Y

1

(t) = Kε₁I₂+ π

3ε₃I₄σ²(t)

K

X

k=1

A_k(t), Y

2

(t) = Kε₁I₃+ π

3ε₃I₅σ²(t)

K

X

k=1

A_k(t),

Yf

0(t) = Kε₁Ie₁+ π

3ε₃Ie₃σ²(t)

K

X

k=1

A_k(t), Yf

1(t) = Kε₁Ie₂+ π

3ε₃Ie₄σ²(t)

K

X

k=1

A_k(t),

with In, eIn defined in (2.5.11).

Denote:

B_k(t, η) =X

l6=k

x_l(t)(φ⁰_l(t) − φ⁰_k(t))G_0,l(t, η), D_k(t, η) =X

l6=k

x_l(t)(φ⁰⁰_l(t) − φ⁰⁰_k(t))G_1,l(t, η), E_k(t, η) =X

l6=k

x_l(t)(φ⁰_l(t) − φ⁰_k(t))G_1,l(t, η), F_k(t, η) =X

l6=k

x_l(t)(φ⁰⁰_l(t) − φ⁰⁰_k(t))G_2,l(t, η)

and

Res₁ = i2πB_k(t, η) + i2πσD_k(t, η) + i2π(η − φ⁰_k(t))res₀− res⁰0

σ(t) − i2πφ⁰⁰_k(t)σ(t)res(2.6.11)₁, Res₂ = 4π²σ(t)E_k(t, η) + 4π²σ²(t)F_k(t) (2.6.12)

+ i2πres₀+ 4π²(η − φ⁰_k(t))σ(t)res₁+ i2πres⁰₁− 4π²φ⁰⁰_k(t)σ²(t)res(2.6.13)₂.

Lemma 2.6.2. [19] Let Res₁ be the quantity defined by (2.6.11). Then,

∂tVex(t, η) =



i2πφ⁰_k(t) − σ⁰(t) σ(t)



Vex(t, η) + i2πφ⁰⁰_k(t)σ(t) eV_x^g1(t, η) − σ⁰(t)

σ(t)) eV_x^g3(t, η) + Res(2.6.14)1

Lemma 2.6.3. [19] For (t, η) satisfying eV_x(t, η) 6= 0 and ∂

∂η(Ve_x^g1(t, η Vex(t, η)) 6= 0, we have

P₀(t, η) = i2πσφ⁰⁰_k(t) + Res₃ (2.6.15)

where

Res₃ = Ve_x(t, η)Res₂− ∂_ηVe_x(t, η)Res₁

Vex(t, η)∂ηVex^g1(t, η) − eVx^g1(t, η)∂ηVex(t, η), (2.6.16) with Res₁ and Res₂ defined by (2.6.11)and (2.6.13) respectively.

Theorem 2.6.4. [19] Suppose x(t) ∈ D⁽²⁾ε1,ε2 with a window function g(t) for some small ε₁, ε₂ > 0. Then, we have the following:

(a) Supposeεe₁ satisfiesεe₁ ≥ ε₀PK

k=1A_k(t) + σ(t)Π₀(t). Then for (t, η) with 



Ve_x(t, η) 

>εe₁, there exists k ∈ {1, 2, ..., K} such that (t, η) ∈ O_k. (b) Suppose (t, η) satisfies |V_x(t, η)| >εe₁, 



∂_η( eV_x^g1(t, η)/ eV_x(t, η)) 

>eε₂, and (t, η) ∈ O_k. Then

ω^apd,2nd,c(t, η) − φx⁰_k(t) = Res₄, (2.6.17)

where

Res₄ = Res₁

i2π eV_x(t, η) − Ve_x^g1(t, η)Res₃

i2π eV_x(t, η) . (2.6.18) Furthermore,

ω^apd,2nd(t, η) − φ⁰_k(t)

< Bd₁ (2.6.19) where

Bd₁ = max

1≤k≤K sup

η∈Ok

n|Res₁|

2πeε₁ + 1 2πεe³₁εe₂

 

Ve_x^g1(t, η)  

  

∂_ηVe_x(t, η) 

|Res₁| +εe₁|Res₂|o

(c) Suppose that ε₁ satisfies the condition in part (a) and Bd₁ ≤ 1 2L_k(t), where

L_k(t) = 1

σ(t)min{α_k+ α_k−1, α_k+ α_k+1}. (2.6.20) Then for any eε3 =εe3(t) > 0 satisfying Bd1 ≤eε3 ≤ 1

2Lk(t),

  lim

λ→0

σ(t) g(0)

Z

|ξ−φ⁰_k(t)|<eε3

R^adp,2nd,λ_x,eε

1,eε2 (t, ξ)dξ − x_k(t)  

≤ Bd₂, (2.6.21) where Bd₂ = Bd⁰₂+ Bd⁰⁰₂ with

Bd⁰₂ = 1

|g(0)|

n

2α_k(εe₁+ σ(t)Π₀(t)) + A_k(t)   

Z

|u|≥α_k

G_k(u)du   +X

l6=k

A_l(t)M_l,k(t) o

,

Bd⁰⁰₂ = 1

|g(0)|{2Π0(t)) + Ak(t) k g k1 |Zt| +X

l6=k

Al(t)Ml,k(t)}

and |Z_t| represents the Lebesgue measure of the set Z_t:

Z_t=n

η : (t, η) ∈ O_k,  

Ve_x(t, η)  

>εe₁, ,  

∂(V_x^g1(t, η)/ eV_x(t, η))  

≤εe₂o .

Chapter 3

Higher-order SST

3.1 High-order Synchrosqueezing transform

3.1.1 The higher-order wavelet synchrosqueezing trans-form

Higher-order WSST

Consider the Nth-order polynomial-phase signal

x(t) = Ae^i2πφ(t) (3.1.1)

with φ(t) = r₁t + ¹₂r₂t²+ · · · + _N¹r_Nt^N, and φ⁰(t) = r₁+ · · · + r_Nt^{N −1}. We have x⁰(t) = i2πφ⁰(t)x(t) and

φ⁰(b + at) = φ⁰(b) +φ⁰⁰(b)at

1! + · · · + φ^{(N )}(b)a^{N −1}t^{N −1}

(N − 1)! . (3.1.2)

We recall the CWT definition of the signal x(t)

W_x(a, b) = Z ∞

−∞

x(b + at)ψ(t)dt. (3.1.3)

Thus, we have

∂_bW_x(a, b) = Z ∞

−∞

x⁰(b + at)ψ(t)dt = Z ∞

−∞

i2πφ⁰(b + a)x(b + at)ψ(t)dt

= Z ∞

−∞

i2π(r₁+ r₂(b + at) + · · · + r_N(b + at)^{N −1})x(b + at)ψ(t)dt

= i2π Z ∞

−∞



φ⁰(b) + φ⁰⁰(b)at

1! + · · · + φ^{(N )}(b)a^{N −1}t^{N −1} (N − 1)!



.x(b + at)ψ(t)dt

= i2π

φ⁰(b)W_x(b, a) +φ⁽²⁾(b)a

1! W_x^ψ1(b, a) + · · · + ^{aN −1}_{(N −1)!}^{φ(N )(b)}Wx^{ψN −1}(b, a) (3.1.4)

where

W_x^ψk(b, a) = Z ∞

−∞

x(b + at)t^kψ(t)dt. (3.1.5)

We can then write _∂b^∂W_x(b, a) =

N

X

k=1

i2πa^k−1

(k − 1)!φ^(k)(b)W_x^ψk−1(b, a).

The goal is to determine φ⁽¹⁾(b) according to W_x(b, a), W_x^ψ1(b, a), . . . , Wx^{ψN −1}(b, a)

∂_bW_x(b, a)

i2πW_x(b, a) = φ⁽¹⁾(b) +

N

X

k=2

a^k−1 (k − 1)!

Wx^ψk−1(b, a)

W_x(b, a) φ^(k)(b). (3.1.6) We can write the equation in the form of a scalar product:

w_x(a, b) = [x₁(b), . . . , x_N(b)][1, V_2,1(b, a), . . . , V_N,1(b, a)]^T,

where w_x(a, b) = ∂_bW_x(b, a)

i2πW_x(b, a) and x_k(b) = φ^(k)(b), for k = 1, . . . , N.

To solve the problem, we pass through successive derivatives of equation

(3.1.6) according to the variable a.

To get sequence x_k(b)

1≤k≤N, we create up a system of N equations with variables x_k(b). Let us denote

y₁ = V_N.X_N^T (3.1.7)

with V_N = [1, V_2,1(b, a), . . . , V_N,1(b, a)] and X_N = [x₁(b), . . . , x_N(b)].

Computing the partial derivatives:

y₂(b, a) = ∂ay₁(b, a)

∂aV2,1(b, a) and V_k,2(b, a) = ∂aV_k,1(b, a)

∂aV2,1(b, a), (3.1.8) which implies the following expression:

y₂(b, a) = [0, 1, V_3,2(b, a), . . . , V_N,2(b, a)]X_N^T (3.1.9)

V_k,2(b, a) = ∂aV_k,1(b, a)

∂aV_2,1(b, a), for k = 3, . . . , N.

To get the j^thequation. We repeat the same process iteratively. We define the new parameter for the AN matrix for j = 2, . . . , N and k = j, . . . , N by:

y_j(b, a) = ∂ay_j−1(b, a)

∂aVj,j−1(b, a), and V_k,j(b, a) = ∂aV_k,j−1(b, a)

∂aVj,j−1(b, a). (3.1.10) Then,

y_j(b, a) = [0, 0, . . . , 1, V_j+1,j, . . . , V_N,j]X_N^T.

We group the N equations and get a good linear system

Since the A_N is an upper triangular matrix with a nonzero diagonal, the solution of the linear system is given by

x_N(b) = y_N(b, a)

We can write this idea in the form of an algorithm.

Determination of the Nth-order local phase transformation

Step 1. We define the matrix AN by (3.1.12) with Vk,jobtained by the following formula:

A_N = Step 3. We solve (3.1.11) and obtain

x_N(b) = y_N(b, a)

Step 4. The Nth-order phase transformation ω_x^N is defined by

ω_x^N(a, b) =

Adaptive higher-order WSST

For a signal with Nth-order polynomial-phase, we define adaptive higher order synchrosqueezing transform fωxN

. We start with the CWT with a time-varying parameter σ(t).

Wf_x(a, b) =

The partial derivative of W (a, b) is given by

We can simplify the expression

∂_bfW_x(a, b) =

In addition for a signal x defined in (3.1.1), eβ(a, b) satisfies

Thus, we have

β(a, b)e

We can also put y₁ in the form of a scalar product defined as follows

Similarly, applying the algorithm (3.1.1) in the case of σ is constant. The Nth-order phase transformation or the reference IF function estimate fωxN

is

3.1.2 The higher-order short time Fourier synchrosqueez-ing transform

Higher-order FSST

Definition 3.1.1. Given a signal x(τ ) = A(τ )e^{i2πφ(τ )} in L²(R) with A(τ ) and φ(τ ) are equal to their Lth-order and N -order respectively, the Taylor expansion for τ close to t (see[17]):

A(τ ) = e^{log(A(τ ))}= expX^L

Since (log(t))^(k)(t) = 0 if L + 1 ≤ K ≤ N , we define the STFT for a signal Applying the derivative of STFT, we have:

∂tV_x^g(t, η) =

The Nth-order IF estimate follows:

Ren

where P_k,1(t, η) = ^Vxtk−1g_Vg ^{(t,τ )}

x(t,τ ) , for k = 2, . . . , N.

Moreover, we can put equation (3.1.5) in the form

w_x(t, η) = [1, P_2,1(t, η), . . . , P_N,1(t, η)]

In the same way, we can present the previous cases to the algorithm 3.1.1 to provide the parameters r₁, . . . , r_N.

Definition 3.1.2. If s ∈ L²(R), the Nth-order local complex IF estimate or phase transformation w^N_x is defined by (see [17]):

w^N_x(t, η) =

Adaptive higher-order FSST

We recall the STFT of the signal x(t) denoted by

Vfx^g(t, η) =

Taking the derivative by t

∂

To simplify the calculation, we note the following expression

Thus, we have

1

Then,

The IF function is obtained by

φ⁽¹⁾(t) = Re in the form of a scalar product previously solved by the algorithm (3.1.1):

we_x(t, η) = [1, S_2,1(t, η), . . . , S_N,1(t, η)]

∂ηS_j,j−1(t, η). In the same way, algorithm (3.1.1) solves the problem.

3.2 New FSST transform

A new phase transformation for the 2nd-order adaptive FSST was proposed in [26]. We consider the new second order FSST associated with STFT. For a signal x(t) defined by

x(t) = Ae^pt+q2t2e^{i2π(ct+12rt2)},

The STFT of x(t) ∈ L₂(R) with a window function g(t) ∈ L2(R) is defined as

V_x(t, η) = Z

R

x(τ )g(τ − t)e−i2πη(τ −t)

dτ (3.2.1)

where t is the time variable and η are is the frequency variable.

As in previous cases, we have

∂_tV_x(t, η) = (p + qt + i2π(c + rt))V_x(t, η) + (q + i2πr)V_x^g1(t, η).

Thus at (t, η) on which V_x(t, η) 6= 0

∂_tV_x(t, η)

Vx^g1(t, η) = p + qt + i2π(c + rt)
Vx(t, η)

Vx^g1(t, η) + (q + i2πr).

Taking partial derivative ∂_η, then we have

∂

∂η

∂_tV_x(t, η) Vx^g1(t, η)

= p + qt + i2π(c + rt)
P₀(t, η) (3.2.2)

where we use P₀(t, η) to denote

P0(t, η) = ∂

∂η

 V_x(t, η) Vx^g1(t, η)



⇒ p + qt + i2π(c + rt)
= 1 P₀(t, η)

∂

∂η

 V_x(t, η) Vx^g1(t, η)

 .

Thus,

c + rt = −p + qt

i2π + 1

i2πP₀(t, η)

∂

∂η

 V_x(t, η) Vx^g1(t, η)

 .

Thus for a general x(t), we define a new phase transformation for the 2nd-order FSST, denoted by ω_x^{N ew,2nd}, as

ω_x^{N ew,2nd}(t, η) =

3.2.1 New higher-order FSST

Definition 3.2.1. Given a signal x(τ ) = A(τ )e^{i2πφ(τ )} in L²(R) with A(τ ) and φ(τ ) are equal to their Lth-order and N -order respectively, the Taylor expansion for τ close to t:

A(τ ) = e^{log(A(τ ))}= expX^L

Applying the derivative of STFT, we have:

∂_tV_x^g(t, η) =

Taking the derivative by η for the equation (3.2.4), we have:

∂_ηw_x(t, η) =(log(A))⁰(t)

Therefore, if in addition, ∂_η

Vx^g(t,η)

where the function w^{N ew}_x is defined by

w_x^{N ew}(t, η) = 1

In addition, we can write equation (3.2.4) in the form

The new version for the Nth-order IF estimate is defined by the STFT:

<n

Moreover, we can put the equation (3.2.4) in the form:

w^{N ew}_x (t, η) = [1, W3,1(t, η), W4,1(t, η), . . . , WN,1(t, η)]

In the same way the previous cases used the algorithm (3.1.1) to provide the parameters r₁, r₃, r₄, . . . , r_N, we can denote k = 3, . . . , N by computing the partial derivatives:

y2(t, η) = ∂_ηω^{N ew}_x (t, η)

∂_ηW_3,1(t, η) and Wk,2(t, η) = ∂_ηW_k,1(t, η)

∂_ηW_3,1(t, η) (3.2.7) which implies the following expression

y₂(t, η) = [0, 1, W_4,2(t, η), . . . , W_N,2(t, η)]R^T_{N −1}. (3.2.8)

To find the j^thequation, we do the same process iteratively. We define the

We group the N − 1 equations and get a good linear system:



Since the AN −1 is an upper triangular matrix with a nonzero diagonal, the solution of the linear system is given by

r_N(t) = y_{N −1}(t, η) Definition 3.2.2. Let x ∈ L²(R). The New version for Nth-order local complex IF estimate or phase transformation ω_x^{N,N ew} is defined by

ω_x^{N,N ew}(t, η) =

Chapter 4

Numerical simulation

4.1 Numerical Simulation

In this section, we present some experimental results for the new second order of the phase transformation ω_x^2nd. Let x(t) be a signal with two linear chirps:

x(t) = x₁(t) + x₂(t) = cos 2π(c₁+ 1

2b₁t)t
+ cos 2π(c₂+1

2b₂t)t
, t ∈ [0, 1]

(4.1.1) where the reference frequencies are c₁ = 12, c₂ = 34, and the chip rates are b1 = 50, b2 = 64. Here, x(t) is sampled uniformly with N = 256 sample points.

We proceed with two representative signal types.

Example 1: Our first signal is a signal with three components, given by s(t) = s₁(t) + s₂(t) + s₃(t),

where

s₁(t) = cos(118π(t −¹₂) + 100π(t −¹₂)²)1_[¹

2,1]. s₂(t) = cos(94πt + 110πt²+ 13 cos(4πt − ^pi₂)).

s3(t) = cos(194πt + 112πt.²)1_[0,³

4].

(4.1.2)

Figure 4.1: The signal s(t) and its components s₁(t), s₂(t) and s₃(t)

We use the relative “root mean square error” (RMSE) to evaluate the separation performance, which is defined by

RM SE = 1 K

K

X

k=1

||z_k− ˆz_k||₂

||z_k|| ,

where ˆz_kis the reconstruction result of z_k, K is the number of components.

Here, we test some parameters of σ from 0.001 to 0.1. The best value for the reconstrucation signal and its components from FSST2 can be obtained by minimizing the RMSE for both approaches of FSST2. Then, σ_Old = σ_{N ew} = 0.023. We colculated the following results.

σ 0.015 0.020 0.023 0.025 0.030 RMSE for Old 0.1375 0.0860 0.0798 0.0843 0.0922 RMSE for New 0.1091 0.0720 0.0717 0.0782 0.0888

Table 4.1: Some differents values of σ and their RMSE.

Figure 4.2: RMSE for FSST2 Old and New with σ ∈ [0.001, 0.04]

As we can see, the best value for minimizing the value of RMSE is σ ≈ 0.023. This value shows the difference between the original IFs and the reconstrucation using the FSST2 old and FSST2 new.

Figure 4.3: The original IF of the signal s(t)

Figure 4.4: Difference of reconstructed IFs with original IFs by old 2nd-order and new 2nd-order FSST.

Figure 4.5: Difference for the reconstructed s₁, s₂, s₃with original component s₁(t), s₂(t), s₃(t) by old 2nd-order and new 2nd-order FSST.

Example 2: The second signal is a signal with two components, given by the signal s(t) = s₁(t) + s₂(t), defined by

s₁(t) = cos(2π(12t + 25t²)) and s₂(t) = cos(2π(34t + 32t²)). (4.1.3)

Figure 4.6: The signal s(t)

Figure 4.7: The components of the signal s(t) one by one s₁(t)(left), and s₂(t)(right)

Here, we test some parameters of σ from 0.001 to 0.1. The best value for the reconstrucation signal and its components from FSST2 can be obtained by minimizing the RMSE for both approaches of FSST2. Then, σOld = σ_{N ew} ≈ 0.05. We colculated the following results.

σ 0.04 0.042 0.045 0.047 0.050

RMSE for Old 0.0893 0.0897 0.0802 0.0827 0.0824 RMSE for New 0.0893 0.0893 0.0800 0.0823 0.0822

Table 4.2: Some Examples for differents parameter of σ.

Figure 4.8: RMSE for Old and New FSST2 with σ ∈ [0.001, 0.1]

As we can see, the best value for minimizing the value of RMSE is σ ≈ 0.045. This value demonstrates the difference between the original IFs and the reconstrucations using the FSST2 old and FSST2 new.

Figure 4.9: The original IF of the signal s(t)

Figure 4.10: Difference of reconstructed IFs with original IFs by old 2nd-order and new 2nd-2nd-order FSST.

Figure 4.11: The difference for the signal s(t)

Figure 4.12: Difference for the reconstructed s₁(t), s₂(t) with original com-ponent s₁(t), s₂(t) by old 2nd-order and new 2nd-order FSST.

Chapter 5

Analysis of Adaptive Shor-time Fourier Transform-based

Synchrosqueezing Transform

5.1 Analysis for new approach of FSST2

The Fourier transform of e^{iπφ00k(t)τ2}g(τ ), which we denote by G_k(ξ), is defined by (refer to [19]):

G_k(ξ) = F

e^{iπ)φ00k(t)τ2}g(τ ) ξ

= Z

R

e^{iπφ00k(t)τ2}g(τ )e^−i2πξτdτ

x_k(t + τ ) = x_k(t)e^{i2π(φ0k(t)τ +12φ00k(t)τ2)}+ A_k(t + τ ) − A_k(t)
e^{i2πφk(t+τ )} + x_k(t)e^{i2π(φ0k(t)τ +12φ00k(t)τ2)} e^{i2π φk(t+τ )−φk(t)−φ0k(t)τ −12φ00k(t)τ2}

− 1
.

Then, we have Vx(t, η) =

K

X

k=1

Z

R

xk(t + τ )g(τ )e^−i2πητdτ

=

K

X

k=1

Z

R

x_k(t)e^{i2π(φ0k(t)τ +12φ00k(t)τ2)}g(τ )e^−i2πητdτ + res₀.

where

res₀ = PK k=1

R

R{ A_k(t + τ ) − A_k(t)
e^{i2πφk(t+τ )}

+ x_k(t)e^{i2π(φ0k(t)τ +12φ00k(t)τ2)} e^{i2π φk(t+τ )−φk(t)−φ0k(t)τ −12φ00k(t)τ2}

− 1
}g(τ )e^−i2πητdτ.(5.1.1)lll

|res₀| ≤Y

0

(t), (5.1.2)

where

Y

0

(t) = Kε₁I₁+ π 3ε₃I₃

K

X

k=1

A_k(t).

We introduce more notations defined as follows

G_j,k(t, η) = Z

R

e^{i2π(φ0k(t)τ +12φ00k(t)τ2)}τ^jg(τ )e^i2πητdτ.

= F



e^{iπ(φ00k(t)τ2)}τ^jg(τ )



η − φ⁰_k(t)
.

We also note

G_k η − φ⁰_k(t)
= G_0,k(t, η) G_j,k = 1

(−i2π)^jG^(j)_k η − φ⁰_k(t)
.

Lemma 5.1.1. [19] Let x(t) = A(t)e^i2πφ(t) ∈ L²(R), we have

∂_tV_x(t, η) = i2πφ⁰_k(t)V_x(t, η) + i2πφ⁰⁰_k(t)V_x^g1(t, η) + Res₁ (5.1.3)

where

Res₁ = i2πB_k(t, η) + i2πD_k(t, η) + i2π(η − φ⁰_k(t))res₀− res⁰₀− i2πφ⁰_k(t)res₁.

Proof. V_x^g0 is defined by (see [19]):

V_x^g0 = i2π

K

X

`=1

x`(t) η − φ<sup>0</sup><sub>`</sub>(t)
G0,`(t, η) − i2π

K

X

`=1

x`(t)φ<sup>00</sup><sub>`</sub>(t)G1,`(t, η) + res⁰₀.

Taking the derivatve by the time variable t, we obtain

∂_tV_x(t, η) = i2πηV_x(t, η) − V_x^g0(t, η). (5.1.4)

Using the derivative of the time variable (5.1.4), we have

∂tVx(t, η) − i2πφ⁰_k(t)Vx(t, η) − i2πφ⁰⁰_k(t)V_x^g1(t, η)

= i2π η − φ⁰_k(t)
V_x(t, η) − V_x^g0(t, η) − i2πφ⁰⁰_k(t)V_x^g1(t, η)

= i2π η − φ⁰_k(t)
V_x(t, η) − i2π

K

X

`=1

x_{`</sub>(t) η − φ<sup>0</sup><sub>`}(t)
G_0,`(t, η) − res⁰₀

+ i2π

K

X

`=1

x_{`</sub>(t)φ<sup>00</sup><sub>`}(t)G_1,`(t, η) − i2πφ⁰⁰_k(t)V_x^g1(t, η).

In addition, we define the expression of V_x(t, η) and V_x^g1(t, η) in the form

Vx(t, η) =

K

X

`=1

x`(t)G0,`(t, η) + res0

V_x^g1(t, η) =

K

X

`=1

x_{`</sub>(t)G<sub>1,`}(t, η) + res₁.

Then, we have

∂_tV_x(t, η) − i2πφ⁰_k(t)V_x(t, η) − i2πφ⁰⁰_k(t)V_x^g1(t, η)

We recall the definition of the new phase transformation for the 2nd-order STFT, which V_x(t, η) 6= 0.:

Then, the new second order complex IF is defined as follows

ω^{N ew,2nd,c}_x (t, η) = P₁(t, η) i2πQ₁(t, η)

where P₁ and Q₁ are defined by

P1(t, η) = ∂

∂η

∂_tV_x(t, η) Vx^g1(t, η)



and Q1(t, η) = ∂

∂η

 V_x(t, η) Vx^g1(t, η)



. (5.1.5) Lemma 5.1.2. For (t, η) such that V_x^g1(t, η) 6= 0 and P₁(t, η) 6= 0, as defined in the equation (5.1.5), we have

P₁(t, η) = i2πφ⁰_k(t)Q₁(t, η) + Res₃ (5.1.6)

and we note that Res₂ = ∂_ηRes₁(see[19]), where

Res₃ = Res₂V_x^g1(t, η) − ∂_ηV_x^g1(t, η)Res₁ Vx^g1(t, η)
2 . Proof. Using the result of lemma (5.1.1), we have

∂_tV_x(t, η) = i2πφ⁰_k(t)V_x(t, η) + i2πφ⁰⁰_k(t)V_x^g1(t, η) + Res₁. (5.1.7)

Therefore, in the equation (5.1.7), therefore, thus

∂tVx(t, η)

Vx^g1(t, η) = i2πφ⁰_k(t) Vx(t, η)

Vx^g1(t, η) + i2πφ⁰⁰_k(t) + Res1

Vx^g1(t, η). (5.1.8) Taking the derivative by η, then yields

∂η

∂tVx(t, η) Vx^g1(t, η)



= i2πφ⁰_k(t)∂η

 Vx(t, η) Vx^g1(t, η)

 + ∂η

 Res1

Vx^g1(t, η)



. (5.1.9)

We replace in the equation (5.1.9) with the expression of the P₁ and Q₁:

P₁(t, η) = i2πφ⁰_k(t)Q₁(t, η) + Res₃. (5.1.10)

This completes the proof of lemma (5.1.2).

Theorem 5.1.3. Let x(t) ∈ D_ε1,ε2 for small ε₁, ε₂ > 0. We have the follow-ing results:

(a) Suppose ε₁ satisfies ε₁ ≥ Π₀(t) + τ₀

K

X

k=1

c_kA_k(t), and for (t, η) with

|V_x^g1(t, η)| > ε₁. Then, there exists k ∈ {1, . . . , K} such that (t, η) ∈ O_k

(b) If (t, η) such that |Vx(t, η)| > ε1,   ∂η

 Vx(t,η) Vx^g1(t,η)

 

 > ε2 and (t, η) ∈ Ok, then, we have

ω^{N ew,2nd,c}_x (t, η) − φ⁰_k(t) = Res₄

where

Res₄ = Res₂V_x^g1(t, η) − ∂_ηV_x^g1(t, η)Res₁ i2π

∂_ηV_x(t, η)Vx^g1(t, η) − ∂_ηVx^g1(t, η)V_x(t, η) . In addition, we have

|ω^{N ew,2nd}(t, η) − φ⁰_k(t)| < Bd₁ where

Bd₁ = max

1≤k≤K sup

η∈Ok

n 1

2πε²₁ε₂ |Res₂||V_x^g1(t, η)| + |Res₁||∂_ηV_x^g1(t, η)|
o

(c) If ε₁ satisfies the condition in part (a) and Bd₁ ≤ ¹₂L_k(t), then

and with |Z_t| represents the Lebesgue measure of the set Z_t:

Z_t:=n

which contradiction the assumption |V_x^g1(t, η)| > ε₁. Therefore, (a) holds.

Proof Part (b). Using the result of Lemma (5.1.2),

P₁(t, η) = i2πφ⁰_k(t)Q₁(t, η) + Res₃. (5.1.12)

Then we have

ω^{N ew,2nd,c}_x (t, η) − φ⁰_k(t) = Res₄

where

Res₄ = Res3

i2πQ₁(t, η).

= V_x^g1(t, η)
2

Res₃ i2π



∂_ηV_x(t, η)Vx^g1(t, η) − ∂_ηVx^g1(t, η)V_x(t, η)

 .

However, the formula of Res₃ is defined in the lemma (5.1.2), and we obtain

V_x^g1(t, η)
2

Res₃ = Res₂V_x^g1(t, η) − ∂_ηV_x^g1(t, η)Res₁.

Then,

Res4 = Res2V_x^g1(t, η) − ∂ηV_x^g1(t, η)Res1

i2π

∂_ηV_x(t, η)Vx^g1(t, η) − ∂_ηVx^g1(t, η)V_x(t, η) .

Next, we have

|Res₄| =  Proof Part (c). First, we have the following result from [7] on p.254

lim

Thus, η ∈ X_t. This which implies the first inclusion Y_t ⊂ X_t.

Using theorem (2.6.4) part(a), if η ∈ Xt,

then |V_x^g1(t, η)| > ε₁ and there exists l ∈1, 2, . . . , K such that (t, η) ∈ O_l. If l 6= k, then

|ω_x,ε^{N ew,2nd}_1,ε2 (t, η) − φ⁰_k(t)| ≥ |φ⁰_k(t) − φ⁰_l(t)| − |φ⁰_l(t) − ω_x,ε^{N ew,2nd}_1,ε2 (t, η)|.

We use the following inequalities

|φ⁰_k(t) − φ⁰_l(t)| > L_k and |φ⁰_l(t) − ω_x,ε^{N ew,2nd}

1,ε2 (t, η)| < Bd₁ ≤ ε₃. Therefore,

 

ω_x,ε^{N ew,2nd}_1,ε2 (t, η) − φ⁰_k(t)  

> L_k(t) − ε₃ ≥ ε₃, and contradicts the assumption η ∈ X_t.

Therefore, l = k and η ∈ Y_t, implying the second inclusion X_t = Y_t. We recall

Z_t:=n

η : (t, η) ∈ O_k,

V_x^g1(t, η) > ε₁,

 

∂_η V_x(t, η) Vx^g1(t, η)

 

≤ ε₂o ,

The fact that X_t = Y_t and Y_t∩ Z_t= φ, with the equation (5.1.13), imply that

Z

Using this equalition

Z For the first term, we get the following results:

T erm₁ = 

substituting the parameters estimated above we get We group all parameter estimates, we find the following results

|

Hence, we have



Then we have the result

  lim

λ→0

1 g(0)

Z

|ξ−φ⁰_k(t)|<ε3

R^adp,2nd,λ_x,ε1,ε2 (t, ξ)dξ − x_k(t)  

≤ Bd₂,

This completes proof of Theorem. 2

Chapter 6

Conclusion and future work

In this study, we introduced a generalization of the STFT-based SST (FSST) with time-varying, CWT-based SST (WSST), the WSST with time-varying, and the FSST with a new phase transformation by using higher order ampli-tude and phase approximations. This generalization allows us to better assess a wide variety of multicomponent signals containing very strongly modulated AM-FM modes.

We also studyed the theoretical analysis of the 2nd-order FSST with a new phase transformation. The new phase transformation is much simpler than the convectional one. The new FSST performance in IF estimation and component recovery is comparable with that of the conventional 2nd-order FSST. In some cses, the new FSST perfermed even better than its conven-tional counterpart.

Since the result showed a better concentration and reconstruction for a wider variety of AM-FM modes, we will continue working on the adaptive FSST with a new phase transformation by using higher order approximations both for the amplitude and phase.

Bibliography

Bibliography

[1] L. Cohen, Time-frequency Analysis, Prentice Hall, New Jersey, (1995).

[2] F. Hlawatsch and G.F. Boudreaux-Bartels, Linear and quadratic time-frequency signal representations,IEEE signal processing magazine., 9.2(1992)21–67.

[3] B. Boashash, Time-frequency signal analysis and processing: a compre-hensive reference, Academic Press., (2015).

[4] P. Flandrin, Time-frequency/Time-scale analysis, Wavelet Analysis and its Applications., (1999).

[5] S. Mallat, A wavelet tour of signal processing, Elsevier., (1999).

[6] I. Daubechies, S. Maes,A nonliner sequeezing of the continuous wavelet transform based on auditory never models. Wavelets in medicine and Biology., (1996) 527-546.

[7] I. Daubechies, J. Lu, and H .-T. Wu, Synechrosqueezed wavelet trans-fonns: An empirical mode decomposition-like tool. Applied and compu-tational harmonic analysis., 30.2 (2011) 243-261.

[8] S. Meignen, T. Oberlin. and S. McLaughlin, A new algorithm for mul-ticomponent signals analysis based on synchrosqueezing: With an ap-plication to signal sampling and denoising. IEEE transactions on Signal Processing., 60.11. (2012 )5787—5798.

[9] G. Thakur and H.-T. Wu, Synechrosqueezing-based recovery of instanta-neous frequency from nonuniform samples, SIAM Journal on Mathemat-ical Analysis., 43.5 (2011) 2078-2095.

[10] T.Oberlin, S. Meignen, and V. Perrier, Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency rep-resentations, IEEE Transactions on Signal Processing., 63.5(2015) 1335-1344.

[11] T. Oberlin and S. Meignen, The second-order wavelet synchrosqueezing transform, 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE., (2017).

[12] H.-T. Wu, Adaptive analysis of complex data sets, Phd. dissertation, Princeton University, Princeton, NJ., (2012)

[13] D. Iatsenko, P.-V. E. McClintock and A. Stefanovska, Linear and syn-chrosqueezed time-frequency representations revisited: Overview, stan-dards of use, resolution, reconstruction, concentration, and algorithms, Digital Signal Processing., 42.7 (2015) 1-26.

[14] L.Li, H.Cai, Q. Jiang, and H. Ji, Adaptive Synchrosqueezing transform with a tim-varying parameter for non-stationary signal separation. Ap-plied and Computational Harmonic Analysis., (2019).

[15] L.Li, H.Cai, H. Han, Q. Jiang, and H. Ji, Adaptive short-time Fourier transform and synchrosqueezing transform for non-stationary signal sep-aration, Signal Processing., 166 (2020)107231.

[16] Y.-L. Sheu, L.-Y. Hsu, P.-T. Chou, and H.-T. Wu, Entropy-based time-varying window width selection for nonlinear-type time-frequency anal-ysis, International Journal of Data Science and Analytics., 3.4 (2017) 231-245.

[17] D.-H. Pham and S. Meignen, High-order synchrosqueezing transform for multicomponent signals analysis - with an application to gravitational-wave signal, IEEE Transactions on Signal Processing., 65.12 (2017) 3168-3178.

[18] R. Behera, S. Meignen, and T. Oberlin, Theoretical analysis of the second-order synchrosqueezing transform, Applied and computational harmonic Analysis., 45.2 (2018) 379–404.

[19] H. Cai, Q. Jiang, L. Li, and Bruce W. Suter, Analysis of adaptive short-time fourier transform-based synchrosqueezing transform, arXiv preprint arXiv.,1812.11033 (2018).

[20] I. Daubechies, Ten Lectures on Wavelets, CBMF Conference Series in Applied Mathematics, Vol. 61 (SIAM, Philadelphia, PA, 1992).

[21] C. L, Herry. M. Frasch,A.J. Seely,and H.-T. Wu, Heart beat classification from single-lead ECG using the synchrosqueezing transform. Physiologi-cal Measurement., 38.2(2017).

[22] C.K. Chui, An Introduction to wavelets, Academic Press., (1992).

[23] Y. Meyer, Wavelets and operators, Volume 1, Cambridge university Press., (1993).

[24] C.K. Chui and Q.T. Jiang, Applied mathematics. Data Compression, Spectral Methods, Fourier Analysis, Wavelets and Applications., (2013).

[25] T. Oberlin, S. Meignen, and V. Perrier, The Fourier-based syn-chrosqueezing transform, 2014 IEEE international conference on acous-tics, speech and signal processing (ICASSP), IEEE., (2014).

[26] Q. Jiang, The second-order synchrosqueezing transform with a new phase transformation, preprint, (2018).

[27] E. Sejdic, I. Djurovic and J. Jiang, Time-frequency feature representa-tion using enery concentrarepresenta-tion: An overview of recent adavances, Digital signal processing., 19.1 (2009) 153–183.

[28] A. Berrian and S. Naoki, Adaptive synchrosqueezing based on a quilted short-time Fourier transform. In Wavelets and Sparsity XVII, Vol. 10394.

International Society for Optics and Photonics., (2017).

[29] L.Li, H.Cai, Z. Wang, Q. Tang, and H. Ji, The Higher-order wavelet synchrosqueezing transform, preprint, (2018).

[30] F. Auger and P. Flandrin, ”Improving the readability of time-frequency and time-scale representations by the reassignment method, IEEE Trans-actions on signal processing., 43.5(1995):1068—1089.

[31] E.Chassande-Mottin, F. Auger, and P. Flandrin, Time-frequency/time-scale reassignment, Wavelets and signal processing, Birkh¨auser, Boston, MA., (2003). 233-267.

[32] N.E. Huang, Z. Shen, S.R. Long, M.L. Wu, H.H. Shih, Q. Zheng, N.C.

Yen, C.C. Tung, and H.H. Liu, The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis.

Proceeding of the Royal Society of London, Series A:mathematical, phys-ical and engineering sciences., 454.1971(1998) 903-995.

[33] P. Flandrin, G. Rilling, and P. Goncalves, Empirical mode decomposi-tion as a filter bank, IEEE signal Processing Letters., 11.2 (2004) 112-114.

[34] Y. Xu, B. Liu, J. Liu, and S. Riemenschneider, Two-dimensional em-pirical mode decomposition by finite elements, Proceedings of the Royal Society London A: Mathematical, Physical and Engineering Sciences., 462.2074 (2006): 3081-3096.

[35] G. Rilling and P. Flandrin, ”One or two frequencies? The empirical mode decomposition answers, IEEE transactions on signal processing., 56.1 (2007) 85-95.

[36] Z. Wu and N.E. Huang, “Ensemble empirical mode decomposition: a noise-assisted data analysis method. Advances in adaptive data analysis., 1.01 (2009) 1-41.

[37] A. Cicone, J.F. Liu, and H.M. Zhou, “Adaptive local iterative filtering for signal decomposition and instantaneous frequency analysis, Applied Computational Harmonic Analysis., 41.2(2016) 384–411.

[38] L. Cohen, Time-frequency distributions-a review,Proceedings of the IEEE., 77.7 (1989) 941-981.

[39] S. Qian and D. Chen, Joint time-frequency analysis, IEEE Signal Pro-cessing Magazine., 16.2(1999) 52–67.

[40] H.-T. Wu, P. Flandrin, and I. Daubechies, One or two frequencies?

The synchrosqueezing answers, Advances in Adaptive Data Analysis., 3.4 (2011) 29-39.

[41] L. Li and H. Ji, Signal feature extraction based on improved EMD method. Measurement., 42.5 (2009) 796-803.

[42] L.Lin, Y. Wang, and H.M. Zhou, “Iterative filtering as an alternative algorithm for empirical mode decomposition, Advances in Adaptive Data Analysis., 1.04 (2009) 543-560.

[43] S. Meignen, D.-H. Pham, and S. McLaughlin, On demodulation, ridge detection and synchrosqueezing for multicomponent signals, IEEE Trans-actions on Signal Processing., 65.8 (2017) 2093-2103.

[44] C. Li and M. Liang, A generalized synchrosqueezing transform for en-hancing signal time-frequency representation, Signal Processing, 92.9 (2012) 2264-2274.

[45] H.Z. Yang, Synchrosqueezed wave packet transforms and diffeomor-phism based spectral analysis for 1D general mode decompositions, Ap-plied and Computational Harmonic Analysis., 39.1 (2015) 33-66.

[46] A.J. Berrian, The Chirped quilted synchrosqueezing transform and its application to bioacoustic signal analysis, Phd. dissertation, University of California at Davis., (2018).

[47] C.K. Chui, Y.-T. Lin, and H.-T. Wu, Real-time dynamics acquisition from irregular samples-with application to anesthesia evaluation, Analysis and Applications., 14.04 (2016) 537-590.

[48] C.K. Chui and H.N. Mhaskar, Signal decomposition and analysis via ex-traction of frequencies, Applied and Computational Harmonic Analysis., 40.1 (2016) 97-136.

[49] C.K. Chui and M.D. van der Walt, Signal analysis via instantaneous frequency estimation of signal components, GEM-International Journal on Geomathematics., 6.1 (2015) 1-42.

[50] Q.T. Jiang and B.W. Suter, Instantaneous frequency estimation based on synchrosqueezing wavelet transform, Signal Processing., 138 (2017) 167-181.

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